Parameter estimation and identifiability, uncertainty, and sensitivity analyses for a COVID-19 model

Implemented model
with
Taken from
Devosmita Sen and Debasis Sen. 2021. Use of a Modified SIRD Model to Analyze COVID-19 Data. Industrial & Engineering Chemistry Research 60 (11), 4251-4260. DOI: 10.1021/acs.iecr.0c04754
Real data
We implemented the number of COVID-19 cases per week reported for Amazonas, a department in Colombia, during the outbreak that occurred in 2020.
load('Amazonas.mat');
ydata = ydata(:, 10:end)';
ydata = [ydata(:,3), ydata(:,2), ydata(:,1)];
D = length(ydata);
Domain = [0 D-1];
xdata = 0 : D-1;
clf
plot(ydata, '-o', 'LineWidth',1.5)
xlim([0 90])
title('Number of cases of COVID-19 in Amazonas-Colombia')
ylabel('Number of cases')
xlabel('Time (days)')
grid on
legend({'Recovered','Dead','Infected'},"Location","best")
Create the model as a table for GSUA-CSB toolbox implementation
First, define the model as a symbolic structure
syms S(t) I(t) R(t) D(t) A(t) E(t) Q(t) C(t)
syms alpha beta delta eta tau epsilon mu gamma k lambda nu z sigma
N = S + C + E + A + I + Q + R + D;
ode1 = diff(S) == -alpha*S - beta*I/N*S - sigma*S*A/N - eta*S;
ode2 = diff(A) == -tau*A + epsilon*E;
ode3 = diff(C) == alpha*S - mu*C;
ode4 = diff(E) == -gamma*E + beta*I/N*S + mu*C + eta*S + sigma*S*A/N - epsilon*E;
ode5 = diff(I) == tau*A + gamma*E - delta*I;
ode6 = diff(R) == lambda*Q;
ode7 = diff(Q) == delta*I - lambda*Q - k*Q;
ode8 = diff(D) == k*Q;
odes = [ode1; ode2; ode3; ode4; ode5; ode6; ode7; ode8];
vars = [R D I S A C E Q];
opts = odeset('NonNegative',1:8);
Then, define the factor intervals with their corresponding names.
Range1 = [ ydata(1,1) ydata(1,1); %R
ydata(1,2) ydata(1,2); %D
ydata(1,3) ydata(1,3); %I
7600 80000; %S
0 2000; %A
0 1000; %C
0 2000; %E
0 2000 %Q
];
Range2 = [ 0 2; %alpha
0 10; %beta
0 1; %delta
0 5; %epsilon
0 1; %eta
0 1; %gamma
0 1; %k
0 1; %lambda
0 1; %mu
0 10; %sigma
0 6; %tau
];
Range = [Range1; Range2];
Finally, use the gsua_dpmat to create the functional table
outs = [1, 3];
% outs = [3];
[T,~] = gsua_dpmat(odes,vars,[0 length(xdata)],'Model','output',outs,'opt',opts,'Range',Range);
Introduce ranges in the following order:
ans(t) = 
T.Row = ["S(0)", "A(0)", "C(0)", "E(0)", "Q(0)",...
"\alpha", "\beta", "\delta", "\epsilon", "\eta", "\gamma", "k", "\lambda", "\mu", "\sigma", "\tau"];
Parameter estimation
First, define the optimization process options: Solver, maximum function evaluation, maximum iterations per estimation, among others.
Solver = 'fmincon';
Opt = optimoptions('fmincon','UseParallel', false,'MaxFunctionEvaluations',30000, ...
'MaxIterations', 3000, 'Display','off');
Define the number of parameter estimations to perform and define the vectors to save the estimated parameters and their respective cost value, i.e., residual.
N = 1000;
Residual = zeros(N,1);
Params = zeros(N,16);
parfor i = 1:N
[T_est, ResTemp] = gsua_pe(T,xdata, ydata(:,[1,3])','solver',Solver,'N',1,'opt',Opt,'save',false);
Residual(i) = ResTemp;
Params(i,:) = T_est.Estfmincon;
end
Sort each set of parameters according to each residual value
[ResidualSorted, idx] = sort(Residual);
EstimatedParams = Params(idx, :);
save("Results/Estimations/Estimations_AmazonasComplex1.mat", "EstimatedParams", "ResidualSorted")
Load the estimated factors and sorted by residual value
addpath('Results')
EstimatedParamsFinal = [];
ResidualFinal = [];
for i = 1:1
load("Results/Estimations/Estimations_AmazonasComplex" + num2str(i) + ".mat");
EstimatedParamsFinal = [EstimatedParams; EstimatedParamsFinal];
ResidualFinal = [ResidualSorted; ResidualFinal];
end
[~, idx] = sort(ResidualFinal);
EstimatedParams = EstimatedParamsFinal(idx, :);
Simulate some factor estimations
We selected 1000-factor estimations to simulate. Those curves follow the trend of real data, the remaining 2000 present aberrant behavior.
N = 1000;
ysim = zeros(N, 2, length(xdata));
parfor i = 1 : N
ysim(i,:,:) = gsua_deval(EstimatedParams(i,:), T, xdata);
end
Starting parallel pool (parpool) using the 'local' profile ... Connected to the parallel pool (number of workers: 8).
Labels = ["Recovered", "Actual cases"];
close all
figure
N2 = 800;
for j = 1:2
subplot(1,2,j)
plot(xdata, squeeze(ysim(1:N2,j,:)), 'b')
hold on
plot(xdata, ydata(:,outs(j)), 'r','Linewidth', 2)
xlim([0 90])
grid on
ylabel(Labels(j))
set(gca,'FontSize',11)
xlabel('Time (days)', 'FontSize', 11)
end
Best factor estimation curve
Best = squeeze(ysim(1,:,:));
BestParams = EstimatedParams(1,:);
% save("Results/Estimations/Best_Amazonas.mat", "BestParams","Best")
load("Results/Estimations/Best_Amazonas.mat")
close all
for j=1:2
subplot(1,2,j)
hold on
plot(xdata, squeeze(ysim(1:N2,j,:)), 'b')
plot(xdata, ydata(:,outs(j)), 'k','Linewidth', 2)
plot(xdata, Best(j,:), 'r','Linewidth', 1.5)
ylabel(Labels(j))
grid on
set(gca,'FontSize',11)
xlabel('Time (days)', 'FontSize', 11)
xlim([0 xdata(end)])
end
Identifiability analysis for 4000-factor estimations
Using the gsua_ia we obtain a table with: a nominal value of each factor as the median of the estimations, the parametric CI using the following equation:
T_ia = gsua_ia(T, EstimatedParams(1:N2,:)', false, false);
T = 16×4 table
 RangeNominalEstindex
1 S(0)3.2330e+043.5641e+043.3986e+043.3988e+044.5592e+044.4736e+043.3147e+045.5906e+043.8791e+046.2730e+043.2504e+043.1225e+043.6551e+043.4866e+042.5927e+042.7367e+042.7827e+042.4050e+042.2945e+042.3094e+042.7671e+045.3967e+042.3109e+042.3122e+042.4316e+042.3884e+042.4538e+042.4306e+042.3906e+043.2503e+043.7662e+043.3766e+042.5173e+046.0957e+042.9454e+046.0672e+043.2748e+043.4525e+045.3528e+044.6510e+041.8257e+042.6516e+044.3610e+043.1650e+044.9476e+043.9243e+043.6136e+044.4250e+045.9177e+041.1997e+042.2450e+045.4081e+044.6815e+045.4902e+042.7056e+041.9819e+046.5894e+043.1862e+043.5356e+043.6906e+043.2636e+045.9344e+041.8663e+045.8984e+041.8321e+045.5239e+045.5387e+046.2957e+041.1150e+042.9111e+041.8784e+042.7884e+042.3450e+041.7812e+041.8488e+041.5953e+043.1531e+043.1579e+042.5395e+043.1975e+042.1215e+042.2728e+042.9529e+041.3443e+042.8901e+044.7834e+042.0489e+042.3650e+041.2873e+043.0654e+041.3696e+043.0889e+044.1989e+041.3965e+042.5243e+041.9537e+043.3989e+041.5860e+043.1328e+042.2073e+044.9461e+041.0650e+043.3644e+042.3801e+042.6573e+043.5014e+044.0815e+041.1813e+041.8159e+049.4463e+031.8190e+043.4920e+042.1577e+042.2051e+044.4707e+042.9046e+041.1648e+042.9463e+043.3426e+042.3280e+041.0364e+044.7415e+041.0653e+043.2043e+041.9813e+042.0892e+043.5318e+041.3544e+042.3043e+045.6329e+043.0334e+048.2759e+031.9826e+041.4333e+043.9903e+043.5684e+045.0288e+043.7922e+042.2061e+041.0344e+041.2082e+042.8107e+041.5158e+041.1700e+042.0902e+045.6035e+044.2964e+041.1675e+042.5610e+042.8648e+041.7977e+041.7623e+041.5415e+041.8385e+047.9002e+035.3695e+041.4123e+042.2138e+044.7501e+043.2046e+041.0456e+041.9647e+042.0662e+041.4172e+041.0153e+043.8270e+041.0483e+045.0151e+041.7801e+043.7559e+044.7906e+045.5327e+045.7504e+043.6221e+044.7670e+043.2512e+041.1932e+043.9157e+045.8192e+041.8183e+041.8955e+042.2841e+043.1269e+042.9757e+049.6811e+036.3811e+043.7910e+042.3275e+044.9757e+044.7324e+045.2612e+041.7592e+043.9342e+047.9987e+042.7654e+041.0578e+046.6429e+044.1070e+042.7431e+045.7890e+043.0677e+043.2490e+042.1971e+042.3740e+042.3133e+041.4245e+042.5404e+042.0928e+042.4152e+043.0672e+042.3743e+042.3965e+045.2779e+045.9346e+043.3540e+043.2173e+042.7227e+048.5416e+032.7305e+042.6351e+045.7198e+044.1341e+046.3352e+045.8941e+044.2549e+041.1651e+041.2521e+044.4525e+045.0934e+046.4764e+044.8048e+044.0850e+043.1528e+044.9770e+045.3958e+047.2504e+045.0456e+049.8756e+031.1796e+044.4232e+045.2341e+048.4343e+035.4252e+044.3108e+042.4856e+045.7898e+041.5961e+045.8253e+047.6004e+032.4514e+046.9970e+046.2064e+041.6370e+041.3233e+045.2302e+041.2651e+043.2969e+041.2378e+045.4068e+044.3974e+046.1510e+045.6307e+043.9165e+043.4191e+043.3194e+046.0114e+041.0380e+046.1208e+044.9750e+041.4118e+043.1618e+049.8805e+037.8823e+034.9465e+041.9603e+041.2881e+042.4938e+049.4611e+034.8789e+049.7178e+034.7469e+041.0896e+044.6810e+042.7003e+042.4699e+047.4085e+043.4493e+043.4863e+045.7155e+046.6985e+043.1601e+045.2640e+041.0046e+045.8413e+042.1406e+044.1798e+043.1360e+043.2502e+043.6067e+046.0993e+043.3911e+043.7098e+046.7113e+043.7927e+041.1447e+043.2309e+046.4152e+044.5921e+042.8614e+047.8421e+043.8094e+045.5248e+045.6483e+047.2086e+046.2889e+047.3905e+044.1678e+046.9672e+042.6973e+047.4720e+047.9465e+044.8354e+041.4464e+042.1212e+045.7080e+041.9553e+046.5813e+045.4780e+046.3975e+046.3068e+042.2933e+041.4147e+045.4912e+042.5863e+046.6507e+045.3389e+041.5203e+042.5717e+041.4074e+041.1235e+044.1172e+041.0975e+045.2074e+046.3331e+047.6014e+035.6486e+045.7243e+046.9363e+041.0603e+042.9919e+041.0951e+046.8422e+046.1562e+041.9414e+045.1018e+043.1172e+048.0000e+041.8048e+046.1659e+045.8312e+045.3555e+043.7868e+046.6540e+041.8711e+046.2840e+042.0321e+042.7161e+049.1120e+032.2854e+041.5951e+041.9019e+044.2129e+041.6351e+043.3558e+044.4947e+041.1109e+047.9879e+034.0287e+046.7176e+042.7433e+047.9950e+047.9999e+043.1748e+043.3035e+043.1364e+043.4344e+047.9998e+041.7482e+044.6978e+042.8265e+045.2608e+045.9596e+041.8462e+044.6886e+042.6422e+044.4234e+045.9457e+044.2479e+046.7168e+044.6498e+047.1397e+043.5891e+047.0088e+046.9290e+042.0619e+047.9974e+043.5186e+041.9674e+045.1144e+043.9315e+046.0464e+043.2181e+043.3252e+045.7249e+044.3715e+046.8659e+046.6620e+046.6407e+042.4907e+045.0554e+044.0439e+042.9966e+047.5279e+047.3517e+043.9928e+046.9093e+046.7909e+043.3468e+045.4150e+045.6644e+046.8705e+045.4389e+047.5447e+042.2568e+042.2555e+042.2649e+042.2588e+042.2576e+042.2568e+042.2503e+042.2577e+042.2519e+042.2572e+042.2761e+042.2633e+042.2313e+042.2326e+042.2586e+042.2422e+042.2439e+041.9930e+046.9780e+042.2504e+042.2157e+048.3812e+031.0846e+042.7246e+041.9399e+043.8615e+042.8425e+042.3094e+044.9991e+044.2678e+045.5938e+043.8992e+043.1804e+041.4413e+044.6804e+042.2314e+042.2087e+042.1925e+042.2101e+042.2350e+042.1857e+042.2207e+042.2246e+042.2258e+042.2238e+042.4367e+042.1984e+042.2076e+042.1876e+042.4986e+043.2733e+043.9929e+042.9637e+042.2276e+046.7134e+042.4524e+041.8508e+047.9958e+036.4119e+043.7508e+044.0684e+044.1395e+043.1990e+043.7071e+043.1626e+047.7286e+034.38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14 \mu0.02060.05610.03840.03840.03840.03840.03840.03840.03840.03840.03840.03830.03830.03840.05400.05350.05350.05410.05410.05480.05440.03840.05410.05380.05410.05530.05420.04770.03820.03830.04990.03830.05200.03840.04680.07110.03840.05560.04180.04060.04560.09270.04480.04350.04780.07390.05780.05800.05810.05400.09480.07010.04880.06680.14360.08420.06920.12720.03350.03080.04950.09460.04020.03630.05650.11970.10730.13820.03520.04010.03980.04140.03970.04160.04030.04070.05360.03960.03760.04000.03980.04150.03850.03960.03920.03990.03890.03950.03950.03930.03980.03940.04170.04160.03630.03890.03900.03950.03990.03870.03990.04070.03740.03750.03810.03770.05620.03870.03870.03840.03990.13440.04060.03740.03390.03910.03870.04020.04230.03580.03890.04000.03790.03340.03590.03720.03590.03920.03650.03780.03580.03810.03600.03740.04210.04010.04040.04020.03520.03870.03800.03660.03600.03740.03630.10300.03620.03870.04130.03770.03680.03920.03630.03460.03660.04100.03850.03430.03690.03500.05130.03800.03830.03900.03750.04300.03620.03360.03760.03490.03430.03550.03840.03160.03860.03550.03560.03750.04360.03520.03950.04010.03340.04100.03690.04100.04190.03910.04560.03290.03950.03630.04162.9640e-080.04090.03487.9132e-090.04260.04040.03570.03720.03370.03520.03940.03930.03530.03950.03890.03960.04020.03880.03880.03564.6554e-080.04050.04610.03050.03580.03820.03970.04210.03246.6929e-082.2004e-080.04080.03340.03730.04130.03275.0423e-040.04150.04000.03470.04150.02743.2996e-060.04150.03700.03220.03760.03980.03570.03290.03630.04680.04160.03780.04170.07550.03670.03716.5464e-040.03450.03040.03560.03653.2970e-060.03690.03610.02860.04862.6511e-040.02523.0298e-042.6311e-072.2740e-070.03480.08750.05450.04330.03270.03190.03920.03000.55100.03450.04810.03600.02720.03621.1887e-050.02910.00100.01360.03023.8158e-040.05510.01440.05380.07680.04270.02210.07940.02580.02830.01370.03030.00370.00290.18380.00270.03461.6366e-060.00890.02190.01080.00860.00230.01645.8487e-090.00680.00970.00129.5751e-040.01060.00140.00240.00540.00970.00250.00330.03914.5972e-070.13630.00590.03020.00110.01690.00800.00140.78740.01147.3602e-040.01580.00730.00860.01090.01840.02421.3370e-060.02120.00170.11170.01610.00125.9077e-040.03240.01720.00640.00120.03030.00150.01220.00760.00160.69303.2198e-050.02910.01630.00250.00180.61730.00180.00850.01282.3853e-070.22160.02210.01190.03550.03380.06040.02440.01898.3467e-040.02250.03140.01160.32852.4479e-040.00350.00100.10612.0586e-040.00270.00418.3235e-044.5056e-040.00540.00630.00140.00130.20281.0569e-040.05120.00190.01310.00920.00760.01370.00590.00560.00126.5380e-040.01322.7507e-040.01880.01330.02730.00200.01291.00000.01900.00630.03861.8103e-060.00871.0427e-040.00611.4469e-040.00590.09859.4208e-059.5695e-058.9203e-041.1447e-070.00100.01750.00494.2369e-040.00302.8246e-060.00420.67870.69030.68600.59390.68300.67270.65970.66080.69590.59960.68150.52300.64950.53010.56380.55210.58930.66430.00160.48240.40080.03210.04960.64550.00782.2284e-040.63890.33290.00760.04510.36120.01040.74480.03860.01080.31890.57320.12590.05950.14065.9384e-060.05940.15160.09140.08660.05460.07710.92710.12180.79020.00200.00770.79970.41350.05000.00190.00170.03451.1881e-040.88860.00150.56840.05530.01320.11630.03130.48020.59190.50450.24330.83040.99490.41460.53320.09560.05630.02026.1081e-040.20617.1023e-040.05560.05880.04200.66690.12090.04103.3246e-040.15900.03130.00750.22670.07790.14370.06320.12720.19390.17810.00320.05720.02054.7509e-070.08830.11530.18320.17070.12363.3103e-040.13240.36060.88610.07130.97400.25730.05130.03620.00130.98290.01230.80000.05290.01570.08110.15460.13010.06206.0920e-060.17820.59670.00120.36550.00810.98320.02450.46250.00770.01640.01360.40282.2702e-040.39943.4991e-040.11870.00170.02970.13290.28370.00540.03880.09772.2803e-043.9552e-040.00100.99800.12350.09710.54760.12180.48230.97330.01109.6236e-040.01500.32310.10100.09421.00004.4495e-050.03110.05400.08890.86470.04080.17260.33080.05153.4762e-040.92360.01740.00480.04211.1716e-040.01020.01140.31440.01384.0030e-041.4915e-040.00330.74320.00590.39711.0271e-040.00340.44150.17340.27150.02860.40770.01190.04760.68960.04970.00720.02080.25370.01070.01930.00140.02070.80781.3241e-090.99500.10210.01370.09140.00830.14670.05506.3223e-071.4533e-050.00154.4792e-060.25900.16140.07155.8253e-110.00200.03260.35290.05630.10510.99920.05690.14720.01130.19891.2691e-070.02700.34410.01200.09040.04230.01790.05380.17094.5876e-040.84180.00760.96621.5949e-077.5801e-040.17110.54630.01650.02810.01040.12117.4255e-040.04960.14460.10670.06670.00660.08771.0468e-060.46640.07431.4729e-040.10378.4684e-090.09200.01135.7451e-050.00441.5743e-050.04510.76210.72194.0015e-040.02920.03540.04110.42010.03580.00940.01730.09350.00474.4718e-060.01870.34170.07950.07280.07620.07432.2363e-100.00180.06850.03400.04740.00820.17560.03770.05950.00250.53850.02410.00990.06810.00390.03550.02750.01850.16450.67524.3946e-050.04130.29418.1270e-080.99990.00290.03610.06680.06230.06530.05150.05970.04330.02630.03780.03970.07100.04320.04810.12340.11060.04311.1837e-110.10183.3119e-111.1564e-113.6793e-110.04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â‹®
Uncertainty analysis for parametric CI
clf
N = 1000;
M = gsua_dmatrix(T_ia,N,'Method','Sobol');
gsua_ua(M, T_ia,'xdata', xdata, 'ynom', ydata(:,outs)', 'parallel', false);
Progress: 100% Estimated processing time (h:m:s): 0:0:2 Remaining time (h:m:s): 0:0:0 Elapsed time (h:m:s): 0:0:2 Estimated stop time (h:m:s): 10:25:33 Number of simulations: 1000
Sensitivity analysis (SA): estimation intervals
Perform the SA using the estimation intervals, i.e., the estimation box
N = 10000;
M = gsua_dmatrix(T,N,'Method','Sobol');
T_sa = gsua_sa(M, T);
Current outputs:1 3 Progress: 6% Estimated processing time (h:m:s): 0:4:31 Remaining time (h:m:s): 0:4:16 Elapsed time (h:m:s): 0:0:15 Estimated stop time (h:m:s): 14:23:45 Number of simulations: 90000 Progress: 12% Estimated processing time (h:m:s): 0:4:29 Remaining time (h:m:s): 0:3:59 Elapsed time (h:m:s): 0:0:29 Estimated stop time (h:m:s): 14:23:43 Number of simulations: 90000 Finding the deepest curve... Progress: 17% Estimated processing time (h:m:s): 0:5:46 Remaining time (h:m:s): 0:4:48 Elapsed time (h:m:s): 0:0:57 Estimated stop time (h:m:s): 14:25:0 Number of simulations: 90000 Progress: 23% Estimated processing time (h:m:s): 0:5:26 Remaining time (h:m:s): 0:4:14 Elapsed time (h:m:s): 0:1:12 Estimated stop time (h:m:s): 14:24:40 Number of simulations: 90000 Progress: 28% Estimated processing time (h:m:s): 0:5:24 Remaining time (h:m:s): 0:3:54 Elapsed time (h:m:s): 0:1:30 Estimated stop time (h:m:s): 14:24:38 Number of simulations: 90000 Progress: 34% Estimated processing time (h:m:s): 0:5:20 Remaining time (h:m:s): 0:3:33 Elapsed time (h:m:s): 0:1:46 Estimated stop time (h:m:s): 14:24:34 Number of simulations: 90000 Progress: 39% Estimated processing time (h:m:s): 0:5:17 Remaining time (h:m:s): 0:3:13 Elapsed time (h:m:s): 0:2:3 Estimated stop time (h:m:s): 14:24:31 Number of simulations: 90000 Progress: 45% Estimated processing time (h:m:s): 0:5:15 Remaining time (h:m:s): 0:2:55 Elapsed time (h:m:s): 0:2:20 Estimated stop time (h:m:s): 14:24:29 Number of simulations: 90000 Progress: 50% Estimated processing time (h:m:s): 0:5:13 Remaining time (h:m:s): 0:2:36 Elapsed time (h:m:s): 0:2:36 Estimated stop time (h:m:s): 14:24:27 Number of simulations: 90000 Progress: 56% Estimated processing time (h:m:s): 0:5:11 Remaining time (h:m:s): 0:2:18 Elapsed time (h:m:s): 0:2:53 Estimated stop time (h:m:s): 14:24:26 Number of simulations: 90000 Progress: 62% Estimated processing time (h:m:s): 0:5:10 Remaining time (h:m:s): 0:2:0 Elapsed time (h:m:s): 0:3:9 Estimated stop time (h:m:s): 14:24:24 Number of simulations: 90000 Progress: 67% Estimated processing time (h:m:s): 0:5:9 Remaining time (h:m:s): 0:1:43 Elapsed time (h:m:s): 0:3:26 Estimated stop time (h:m:s): 14:24:24 Number of simulations: 90000 Progress: 73% Estimated processing time (h:m:s): 0:5:11 Remaining time (h:m:s): 0:1:26 Elapsed time (h:m:s): 0:3:44 Estimated stop time (h:m:s): 14:24:25 Number of simulations: 90000 Progress: 78% Estimated processing time (h:m:s): 0:5:11 Remaining time (h:m:s): 0:1:9 Elapsed time (h:m:s): 0:4:2 Estimated stop time (h:m:s): 14:24:26 Number of simulations: 90000 Progress: 84% Estimated processing time (h:m:s): 0:5:10 Remaining time (h:m:s): 0:0:51 Elapsed time (h:m:s): 0:4:19 Estimated stop time (h:m:s): 14:24:24 Number of simulations: 90000
sum(T_sa.Si)/sum(abs(T_sa.Si))
ans = 0.9339
clf
gsua_plot('Pie',T_sa,T_sa.STi)
SA: parametric confident intervals
Perform the SA using the confident intervals using the median value of the parameters
N = 10000;
M = gsua_dmatrix(T_ia,N,'Method','Sobol');
T_sa = gsua_sa(M, T_ia);
Progress: 7% Estimated processing time (h:m:s): 0:1:22 Remaining time (h:m:s): 0:1:16 Elapsed time (h:m:s): 0:0:5 Estimated stop time (h:m:s): 17:57:5 Number of simulations: 75000 Progress: 14% Estimated processing time (h:m:s): 0:1:21 Remaining time (h:m:s): 0:1:10 Elapsed time (h:m:s): 0:0:10 Estimated stop time (h:m:s): 17:57:4 Number of simulations: 75000 Progress: 20% Estimated processing time (h:m:s): 0:1:22 Remaining time (h:m:s): 0:1:5 Elapsed time (h:m:s): 0:0:16 Estimated stop time (h:m:s): 17:57:5 Number of simulations: 75000 Progress: 27% Estimated processing time (h:m:s): 0:1:22 Remaining time (h:m:s): 0:1:0 Elapsed time (h:m:s): 0:0:22 Estimated stop time (h:m:s): 17:57:5 Number of simulations: 75000 Progress: 34% Estimated processing time (h:m:s): 0:1:23 Remaining time (h:m:s): 0:0:55 Elapsed time (h:m:s): 0:0:27 Estimated stop time (h:m:s): 17:57:6 Number of simulations: 75000 Progress: 40% Estimated processing time (h:m:s): 0:1:24 Remaining time (h:m:s): 0:0:50 Elapsed time (h:m:s): 0:0:33 Estimated stop time (h:m:s): 17:57:7 Number of simulations: 75000 Progress: 47% Estimated processing time (h:m:s): 0:1:24 Remaining time (h:m:s): 0:0:45 Elapsed time (h:m:s): 0:0:39 Estimated stop time (h:m:s): 17:57:7 Number of simulations: 75000 Progress: 54% Estimated processing time (h:m:s): 0:1:25 Remaining time (h:m:s): 0:0:40 Elapsed time (h:m:s): 0:0:45 Estimated stop time (h:m:s): 17:57:9 Number of simulations: 75000 Progress: 60% Estimated processing time (h:m:s): 0:1:27 Remaining time (h:m:s): 0:0:34 Elapsed time (h:m:s): 0:0:52 Estimated stop time (h:m:s): 17:57:10 Number of simulations: 75000 Progress: 67% Estimated processing time (h:m:s): 0:1:27 Remaining time (h:m:s): 0:0:29 Elapsed time (h:m:s): 0:0:58 Estimated stop time (h:m:s): 17:57:10 Number of simulations: 75000 Progress: 74% Estimated processing time (h:m:s): 0:1:27 Remaining time (h:m:s): 0:0:23 Elapsed time (h:m:s): 0:1:4 Estimated stop time (h:m:s): 17:57:10 Number of simulations: 75000 Progress: 81% Estimated processing time (h:m:s): 0:1:27 Remaining time (h:m:s): 0:0:17 Elapsed time (h:m:s): 0:1:10 Estimated stop time (h:m:s): 17:57:10 Number of simulations: 75000 Progress: 87% Estimated processing time (h:m:s): 0:1:27 Remaining time (h:m:s): 0:0:11 Elapsed time (h:m:s): 0:1:15 Estimated stop time (h:m:s): 17:57:10 Number of simulations: 75000 Progress: 94% Estimated processing time (h:m:s): 0:1:27 Remaining time (h:m:s): 0:0:5 Elapsed time (h:m:s): 0:1:21 Estimated stop time (h:m:s): 17:57:10 Number of simulations: 75000 Progress: 100% Estimated processing time (h:m:s): 0:1:27 Remaining time (h:m:s): 0:0:0 Elapsed time (h:m:s): 0:1:27 Estimated stop time (h:m:s): 17:57:10 Number of simulations: 75000
sum(T_sa.Si)/sum(abs(T_sa.Si))
ans = 0.9996
clf
gsua_plot('Pie',T_sa,T_sa.STi)